8 Rev. J. Challis on the theoretical Determination 



nication to the Phil. Mag. and Annals of Philosophy for 

 August 1829, which contained several inaccuracies, I made 

 an assertion respecting this case of fluid motion, the correct- 

 ness of which subsequent consideration has only tended to con- 

 firm ; viz. that when udx + vdy + w d z is an exact differential, 

 the whole motion is such that the motion of each elementary 

 portion of the fluid is directed to a fixed or moveable centre. 

 The course of the reasoning by which this proposition may 

 be established is I conceive such as follows. We know from 

 the theory of partial differential equations, that their integrals, 

 whether we can obtain them exactly or not, must contain arbi- 

 trary functions. The arbitrariness of which we are informed 

 by pure analysis, has a signification in the applications of the 

 functions to physical questions. Thus the existence of arbitrary 

 functions in the integrals of the equations which determine the 

 motion of fluids, is the proper proof that we can give to the 

 fluid any motion we please ; and this is an evident consequence 

 of one of the fundamental principles in the investigation of 

 the motion, the perfect mobility of the particles. The forms 

 of the functions depend on the particular motion we choose to 

 give to the fluid by vessels, pipes, or other means. But how- 

 ever irregular we may cause the motion to be, it may still be 

 conceived to be composed of elementary motions, which obey 

 the law of continuity, independently of our will, just as a line, 

 however broken and irregular, may be conceived to be made 

 up of elementary portions which are straight lines. Absolute 

 discontinuity is inconceivable. The law of these motions will 

 be independent of time and position, and dependent only on 

 the nature of the fluid. Hence, to learn whether the motion 

 be really so composed, it will be necessary, after having ob- 

 tained the complete integral of the equation expressing the 

 continuity of the fluid, to ascertain whether the arbitrary func- 

 tions which the integral contains, can be shown to have a 

 particular form, when discussed on the supposition that the 

 origins of the time and coordinates are not fixed. This will 

 in general require the solution of a functional equation. An 

 instance of this reasoning was given in the communication 

 above mentioned, for the case in which the motion is in space 



of two dimensions. From the complete integral of ^- -f 



ds? dy* 



0, a particular form of the arbitrary functions was obtained, 

 which indicated that the velocity was directed to a centre and 

 varied inversely as the distance from the centre. I have since 



found that the complete integral of -~- + ~- + -~f = 0, 



which M. Poisson has expressed by definite integrals (Mem. de 



VAcad. 



